# stroke sequence

Sat Dec 27 11:04:48 CET 2003

```    Hello, seqfans
I posted two sequences to OEIS.

%I A002620
%S A002620 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, ....

%N A002620 A(n) gives possible maximal numbers of strokes on  K_n.
Where all edges on K_n have directions.
A "stroke" is defined as follows :
A local maximal di-path on di- graph.

%e A002620
n=3 , maximal two strokes exist, "x -> y -> z" and " x -> z" , so
A(3)=2 .
n=4, maximal four strokes exist, "u -> x -> z" and "u -> y" and "u -> z"
and "x -> y -> z" , so A(4)=4 .
%Y A002620 Cf.A000001
%K A002620 nonn
%O A002620 1,3
%A A002620 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)

I asked Neil  to add some of this description on comment line of
a002620.

%I A000001
%S A000001 0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ....
%N A000001    B(n) gives minimal numbers of strokes on  K_n.
Where all edges on K_n have directions.
A "stroke" is defined as follows :
A local maximal di-path on di- graph.
%C A000001    If B(n)=1 then K_n has an Euler path.
%Y A000001 Cf. A002620
%K A000001 nonn
%O A000001 1,4
%A A000001 Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)

This definition of "stroke" has still some ambiguousness.
At least, a condition that different two strokes don't have the same
edges is needed.
Even if the condition is satisfied, there are some ambiguousness.
For example : a di-graph "H"
a->b->c->d,
x->b, c->y
Four sets of strokes are able to exist.
{a->b->c->d, x->b, c->y}
{a->b->c->y, x->b, c->d}
{x->b->c->d, a->b, c->y}
{x->b->c->y, a->b, c->d}
But, after all, the number of strokes is the same.
So, the number doesn't depend on these sets.

Do you understand what I am considering about?

Yasutoshi

```